Let $f:B_r(x_0)\subset \mathbb{R}^n\to\mathbb{R}^n$ a $C^1$ diffeomorphism with $\Vert f'(x)^{-1}\Vert \le M$ for all $x\in B_r (x_0)$ and $\vert f(x_0)\vert <\frac{r}{M}$.
I'm wondering if I can say that the set $f(B_r(x_0))$ is convex. I was thinking it could be true, but I didn't find this result, so I was in doubt, but I can't demonstrate the result or find a counterexample.
Does anyone have any ideas?
The ideas I had, guarantee this result only locally, for example, being locally lipschitz and locally uniformly continuous, but I didn't think of a way to guarantee this globally.